The case for True Length = Rest Length

[JesseM]

This presupposes that "having a velocity" has any absolute meaning.

I didn't mean to suggest that, obviously velocity is coordinate-dependent, but there is certainly a clear procedure for measuring velocity relative to any given inertial frame.

If twin B is the traveler that later returns, we could equally analyze the situation by saying that A was the initial traveler, and B greatly accelerated to catch up with him at a later point. Mathematically, you will arrive at the same answer.

Of course, you can evaluate the integral using v(t), t0 and t1 from different frames, and you'll always get the same answer for the elapsed proper time.

Ignoring gravity, the existence of an objective age differential in the Twins Paradox is caused by acceleration while its magnitude is determined by their relative velocities.

Yes, I like that way of stating it.

 

[ghwellsjr]

Ignoring gravity, the existence of an objective age differential in the Twins Paradox is caused by acceleration while its magnitude is determined by their relative velocities.

Yes, I like that way of stating it.

Not me. Relative velocities are caused by accelerations. The magnitude of the age difference is determined by their relative velocities and the time interval over which those velocities exist.

Acceleration is required to bring the two twins back to the same location so that when the clocks are compared in any frame of reference, they will have the same age difference as they would in any other frame of reference. But in and of itself, the acceleration has nothing to do with the age difference.

 

[rjbeery]

I would like you to come to an understanding that your proof is defective, reciprocal time dilation (and length contraction) are not nonsensical, and they don't lead to any logical contradiction.

Objectively, I'm taller than you AND you're taller than me. This is a nonsensical statement which contains a logical contradiction.

Can you prove an age differential exists if only one twin accelerates but never comes back to the first twin? For example, the traveling twin accelerates away from the first twin. Does that mean he's younger? After awhile the traveling twin decelerates so that he is at rest with respect to his brother but far away. Does that mean he's younger? The traveling twin now accelerates back toward his brother. Does that mean he's younger? The traveling twin now decelerates when he's only half way back so that he is once more at rest with respect to his brother but at half the distance than the first time he stopped. Does that mean he's younger?

Although I've said many times that a reunion is necessary to objectively determine anything, I'm now wondering if this is necessarily the case. If the accelerating twin ever visually sees the other's clock as being ahead of his own then he can conclude objectively that he is younger. I believe in all scenarios given this event occurs prior to their actual reunion. Whether or not this event occurs in your above situations depends on the particulars of your various scenarios.

You will find in there an explanation of the Twin Paradox that does not involve acceleration at all. It's on the second page, second column. It references a similar graph you put in your post and advised me to read for more analysis

That paper was referenced to explain my graph. I don't necessarily agree with the author's conclusions on everything. However, on the point you raised, he concludes with

Therefore, on outgoing and returning legs both traveling and stationary clocks seem to be going faster than each other, but the change of inertial frames at e constitutes a change of lines of simultaneity which results in a jump ahead between the times r and s as measured on the moving clocks with respect to the stationary clocks. The "missing time" between r and s becomes then the REASON for the differential aging.

Again, even with the 3 brothers version of the twin paradox, it isn't the relative velocity that causes the age differential; on both the outgoing and incoming legs the time dilation is reciprocal. It's in the 2 traveling brothers' change of inertial frames that "causes" what he calls the "missing time". Even though his conclusion is that it is STILL NOT VELOCITY (or speed) that caused the age differential, and therefore does not help your argument, this methodology of avoiding acceleration by passing information between moving frames makes me raise an eyebrow.

 

[rjbeery]

But in and of itself, the acceleration has nothing to do with the age difference.

Yes, except the existence of it.

 

[JesseM]

Not me. Relative velocities are caused by accelerations. The magnitude of the age difference is determined by their relative velocities and the time interval over which those velocities exist.

But presumably you'd agree that if I evaluate the integral \int_{t_0}^{t_1} \sqrt{1 - v(t)^2/c^2} \, dt for two paths between the same two points in spacetime, as a general rule if one path has a constant v(t) while the other has a varying v(t), that fact alone is enough to guarantee the one with the varying v(t) will have a smaller value when the integral is evaluated, without knowing any additional information about the specifics of the problem?

Acceleration is required to bring the two twins back to the same location so that when the clocks are compared in any frame of reference, they will have the same age difference as they would in any other frame of reference. But in and of itself, the acceleration has nothing to do with the age difference.

Suppose we draw two paths on a 2D plane between a pair of points A and B, one of them is a straight line while the other consists of two straight segments at different angles connected by a curved segment. Since we know a straight line is the shortest distance between points, we know the one with the curved segment will have a longer total length. Would you say that "in and of itself, the fact that one path has a curved segment has nothing to do with the difference in length"? If so I guess I don't know what you mean by "in and of itself" (and if you wouldn't say that, you'd be missing how closely statements about elapsed time in spacetime are analogous to statements about path length in space, see [post=2972720]this post of mine on the details of the geometric analogy[/post]).

 

[Dale]

Geometrically, attributing the difference in aging to the acceleration is the same as attributing the fact that the sum of two sides of a triangle are longer than the remaining side to the vertex between the two sides. While it is true that the vertex unambiguously occurs on the longer path the vertex itself has no length so it cannot be said to be the source of the extra length. The extra length is in the sides, not the vertex.

 

[rjbeery]

In other words...

geometrically, attributing the difference in aging to the acceleration is the same as attributing the fact that the sum of two sides of a triangle are longer than the remaining side to the vertex between the two sides. While it is true that the vertex unambiguously occurs on the longer path the vertex itself has no length so it cannot be said to be the source of the extra length. The extra length is in the sides, not the vertex.

=

ignoring gravity, the existence of an objective age differential in the twins paradox is caused by acceleration while its magnitude is determined by their relative velocities.

 

[JesseM]

Perhaps it's the ambiguous meaning of the phrase "caused by" that's the source of disagreement, you could instead say something like "the fact that twin B accelerated is a necessary and sufficient condition for him to have aged less than the inertial twin A when they reunite".

 

[rjbeery]

I would be amenable to that, reiterating that we're ignoring gravity.

 

[ghwellsjr]

OK rjbeery, let's see if we can find some common ground.

Do you agree with this statement:

When a clock accelerates, its tick rate changes.​

 

[JesseM]

OK rjbeery, let's see if we can find some common ground.

Do you agree with this statement:

When a clock accelerates, its tick rate changes.​

Again coordinate-dependent, in a non-inertial coordinate system a clock can be experiencing proper acceleration (or coordinate acceleration) but having an unchanging tick rate. Would be better to say "when a clock accelerates, its tick rate changes relative to all inertial frames."

 

[rjbeery]

If we're trying to speak strictly in objective terms then I am unable to agree with that statement. The holder of the clock would say no while all others would say yes.

 

[DrGreg]

Would be better to say "when a clock accelerates, its tick rate changes relative to all inertial frames."

Actually, even that statement need not be true. A clock moving at constant speed round a circular path is accelerating, but relative to someone permanently at the centre of the circle, the clock rate is unchanging!

How about "when a clock properly accelerates in a straight line, its tick rate changes relative to all inertial frames."?

 

[bobc2]

Ignoring gravity, the existence of an objective age differential in the Twins Paradox is caused by acceleration while its magnitude is determined by their relative velocities.

 

[rjbeery]

How about "when a clock properly accelerates in a straight line, its tick rate changes relative to all inertial frames."?

Agreed.

A clock moving at constant speed round a circular path is accelerating, but relative to someone permanently at the centre of the circle, the clock rate is unchanging!

I like this! Of course, it also occurs to me that if Twin B circles Twin A he will clearly age less but, because their relative speed is zero, it's another counter-example to ghwellsjr's statement below.

And I didn't claim that it is relative velocity that leads to an age difference. I said it is a relative speed over a period of time that leads to an age difference.

 

[rjbeery]

bobc2, in the Minkowski diagrams how does an object do such things as create a wavy timeline or rotate its timeline by 90 degrees so that it may "take a shortcut"?

 

[DrGreg]

Of course, it also occurs to me that if Twin B circles Twin A he will clearly age less but, because their relative speed is zero, it's another counter-example to ghwellsjr's statement below.

Er, no, because "speed" means scalar magnitude of vector velocity, not rate of change of scalar separation. The relative speed is not zero in this case.

 

[rjbeery]

Er, no, because "speed" means scalar magnitude of vector velocity, not rate of change of scalar separation. The relative speed is not zero in this case.

Maybe I was hasty; I was considering relative speed to be change in relative distance over time...let's try another tack: Twin A is the one with relative speed from the orbiting Twin B's perspective, yet they both agree that B is aging more slowly...does this suggest that it isn't speed but acceleration that is the cause?

 

[bobc2]

bobc2, in the Minkowski diagrams how does an object do such things as create a wavy timeline or rotate its timeline by 90 degrees so that it may "take a shortcut"?

Answer: The same way it does in the 3-D world race between blue and red cars (my post #224). The blue car accelerated to turn the corner for the short cut. The blue guy traveling twin in the 4-D twin paradox example did the same thing.

That's the whole point. Sure, you need the acceleration to rotate in 4-D space, but that doesn't mean the acceleration caused the path to be shorter, any more than the acceleration for the blue 3-D race car "caused" the short cut (even though you needed to accelerate in order to take the shorter path).

Now, I'm beginning to see the source of our difference in preferred way of interpreting the twin paradox. You like to stick close to the phenomenological approach--just concentrate on the observations and don't make too much of ideas that are not solidly supported by well identified postulates and theorems (although I'm not convinced that even from that point of view you are on solid ground with your acceleration causality idea--but I'll let the others sort that out). My problem is I tend to come down on the side of the realists and would like to have a "real" (whatever that means) objective 4-dimensional space with 4-D objects (it has serious implications about the interpretation of causality--and I certainly would not accept the idea that acceleration "causes" the affine space connection to the laws of physics. I tend to want physics to be expressed geometrically. However I am ambivalent about that as well, because I know quite well the implications that follow--and at a subjective level I don't like those implications at all. (I'm sure Ben Crowell and JesseM would have some pretty tough questions for those green men from hyperspace in the sketch below.)

Wow! I've got to hand it to you, RJBeery. You sure have your hands full and are doing a gallant job of fending off everyone. Stick to your guns.

 

[JesseM]

Answer: The same way it does in the 3-D world race between blue and red cars (my post #224). The blue car accelerated to turn the corner for the short cut. The blue guy traveling twin in the 4-D twin paradox example did the same thing.

That's the whole point. Sure, you need the acceleration to rotate in 4-D space, but that doesn't mean the acceleration caused the path to be shorter, any more than the acceleration for the blue 3-D race car "caused" the short cut (even though you needed to accelerate in order to take the shorter path).

Wow! I've got to hand it to you, RJBeery. You sure have your hands full and are going a gallant job of fending off everyone.

This whole argument seems to turn on semantics--how are you defining "caused"? I don't think it's a word that has any formal technical definition in physics. Presumably there are situations where X and Y are two physical facts, and X is a necessary and sufficient condition for Y, but you would not agree that "X caused Y"?

 

[Dale]

In other words...

=

No. I think it is weird to talk of length being "caused" by a bend. Even with the caveat about the magnitude.

 

[harrylin]

Maybe I was hasty; I was considering relative speed to be change in relative distance over time...let's try another tack: Twin A is the one with relative speed from the orbiting Twin B's perspective, yet they both agree that B is aging more slowly...does this suggest that it isn't speed but acceleration that is the cause?

The cause of what? If you check out the equations, time dilation is only a function of the velocity - more precisely, of the velocity wrt to any inertial reference system. For the Twin case, we are dealing with two velocities and acceleration is the cause of the asymmetry, by changing one of the velocities. This was understood from the very start when the example was given.

The time dilation equation implies that clock rate is not affected by acceleration, and this is correct for many clocks. Tests with accelerator rings have confirmed this for elementary particles. See "The Clock Hypothesis" in http://www.phys.ncku.edu.tw/mirrors/physicsfaq/Relativity/SR/experiments.html

Harald

 

[bobc2]

This whole argument seems to turn on semantics--how are you defining "caused"? I don't think it's a word that has any formal technical definition in physics. Presumably there are situations where X and Y are two physical facts, and X is a necessary and sufficient condition for Y, but you would not agree that "X caused Y"?

You always seem to come up with the relevant observations, JesseM. I don't think I could disagree with you. Also, I originally thought RJBerry's posts were a little off the wall, but I've gained much more respect for his ideas (not that what I think about his ideas is relevant).

 

[rjbeery]

This whole argument seems to turn on semantics--how are you defining "caused"?

I've been thinking about the semantics issue and how it may be resolved...if we look at the logical definition of causality, there are 3 categories of causes:

Necessary causes:
If x is a necessary cause of y, then the presence of y necessarily implies the presence of x. The presence of x, however, does not imply that y will occur.
Sufficient causes:
If x is a sufficient cause of y, then the presence of x necessarily implies the presence of y. However, another cause z may alternatively cause y. Thus the presence of y does not imply the presence of x.
Contributory causes:
A cause may be classified as a "contributory cause," if the presumed cause precedes the effect, and altering the cause alters the effect. It does not require that all those subjects which possesses the contributory cause experience the effect. It does not require that all those subjects which are free of the contributory cause be free of the effect. In other words, a contributory cause may be neither necessary nor sufficient but it must be contributory.

Using these definitions we can begin labeling things.

IGNORING GRAVITY: Relative velocity, on its own, is necessary but not sufficient for absolute time dilation. Acceleration, on its own, is also necessary but not sufficient. Only taken together, and only under certain circumstances, are velocity AND acceleration necessary and sufficient for absolute time dilation. (How do we account for the "only under certain circumstances"? I'm thinking of the times in which both twins experience a combination of velocity and acceleration such that the time dilation effects are nullified.)

WITH GRAVITY: Relative velocity is now neither necessary nor sufficient for absolute time dilation, but now falls under the contributory cause definition. Acceleration WITH Relative velocity, as above, continues to be sufficient. Gravity, on its own, is also sufficient (again, "only under certain circumstances"). There is also what I consider to be a dubious example of the twin paradox involving 3 brothers that is devoid of either acceleration or gravity which can be found http://chaos.swarthmore.edu/courses/PDG/AJP000384.pdf". The author concludes that the "cause" of time dilation here is the change of inertial frames, which I should note, also occur under the other two mentioned sufficient causes. Presuming this encompasses ALL possible sufficient causes for absolute time dilation, we have now identified a single globally valid necessary cause: a change in inertial frames for at least one of the twins.

 

[GrayGhost]

JesseM,

Sorry, my PC has been choking a bit as late. I'll review your prior recent posts again before responding. My prior illustration was initially drafted for the all-inertial case, and I tweeked it (quickly) for the twin B POV only for sake of point. Attached is an enhancement of the prior illustration in an attempt to resolve your complaints about it.

I designate numbers (0 thru 8) on the illustration, which should not be taken as "an order of events". They are merely to have references in any discussion of the illustration.

When twin B executes a virtually-instant-acceleration, he must hold (per the LTs) that twin A moves rapidly from point 1 to point 3. The twin A clock at point 1 is (virtually) the same time readout as at point 2. Therefore, during B's rapid acceleration, the A clock must wildly advance 75% of the A-time from point 2 to point 4 (given v = 0.866c). Doppler effects do not directly show a "time jump", however indirectly they do after the doppler effects are accounted for and light transit time (from twin A) is negated.

GrayGhost

 

[JesseM]

Sorry, my PC has been choking a bit as late. I'll review your prior recent posts again before responding.

OK, I hope you will pay particular attention to the end of my previous post to you, and answer the questions I asked you there.

When twin B executes a virtually-instant-acceleration, he must hold (per the LTs) that twin A moves rapidly from point 1 to point 3.

No, because there is no single inertial frame where such a rapid jump occurs. Of course it's true that if you use the LT to find the distance from A to B immediately before acceleration in the inertial frame where B was at rest before acceleration, and then use the LT to find the distance from A to B immediately after the acceleration in the frame where B was at rest after the acceleration, the two distances are different. But each of these two distinct frames cover all of spacetime, both the times before acceleration and the times after, and in neither frame does A's position make any sudden jump. It's only if you make a Frankenstein's monster frame by stitching together the first frame's coordinates for events before the acceleration and the second frame's coordinates for events after (i.e., make a non-inertial frame whose judgments about simultaneity and distance at each point on B's worldline matches with B's instantaneous inertial rest frame at that point) that you can say that A's position "jumped" as in your diagram. But the Lorentz transformation does not deal with Frankenstein's monster frames created by stitching together regions of different inertial frames, it only deals with mappings between one inertial frame and another inertial frame. If you use the LT to map from the x,t coordinates in A's frame to some set of x',t' coordinates in any other frame, you'll never come up with a single coordinate system where the x'-coordinate of A jumped by a large amount over an arbitrarily brief interval of coordinate time dt'.

 

[GrayGhost]

No, because there is no single inertial frame where such a rapid jump occurs.

Of course, but it must occur per he who transitions said inertial frames. He is B. Twin B is actively transitioning inertial frames of reference. It seems to me that the LTs still tell us what should happen, by extrapolation ...

We may consider a number of inertial observers all momentarily colocated at twin B's departure event from twin A. One at 0.1c, another at 0.2c, 0.3c, etc thru 0.866c. If B's acceleration is virtually instant, virutally no time passes for he or anyone else momentarily colocated at the departure event. Twin B transitions the frames in almost no time at all, and his final frame is 0.866c. In the limit as acceleration approaches instant, at completeion of B's acceleration, twin B and the always inertial 0.866c observer are essensitally one in the same far as their POV goes. When he completes the rapid acceleration, he should record (just about) the same of the planet X clock that the always inertial 0.866c observer does.

The Fermi Walker cooridnates appear to do the very same thing. Look at the animation of the right side of this link ... and note the wild jumps during proper acceleration ...

http://en.wikipedia.org/wiki/Fermi%E2%80%93Walker_transport"

GrayGhost

 

[JesseM]

Of course, but it must occur per he who transitions said inertial frames. He is B. Twin B is actively transitioning inertial frames of reference.

What does "transitions inertial frames of reference" mean? You seem to have some totally over-concrete notion that objects are just naturally "in" some frame or another, as opposed to it being a mere human convention to link particular objects to particular frames. Nothing is stopping B from continuing to use his old inertial rest frame (or any other frame) even when he is no longer at rest in it, there is no reason for him to "transition" unless he chooses to do so!

Twin B transitions the frames in almost no time at all, and his final frame is 0.866c. In the limit as acceleration approaches instant, at completeion of B's acceleration, twin B and the always inertial 0.866c observer are essensitally one in the same far as their POV goes. When he completes the rapid acceleration, he should record (just about) the same of the planet X clock that the always inertial 0.866c observer does.

Again, the coordinates you "record" for distant objects and events (including the coordinate distance to a distant planet) is a matter of what frame you choose to use, it's not like you are "naturally" forced to record things in the inertial reference frame where you are currently at rest. You don't seem to understand this point about the frame associated with a given observer being purely a matter of choice/convention as opposed to something natural, which is exactly why I asked you the following question in post #201, then repeated it in #204, then again asked you to address it in post #236:

Objects don't have a "sense-of-simultaneity", again it is simply a matter of convention what coordinate system we associate with what object. As I said, even if I am an inertial observer I am perfectly free to use an inertial coordinate system moving at 0.6c relative to me, this goes against the most common convention for what is meant by the words "my perspective" but as long as I explain what I'm doing there is no physical reason why I am "wrong" to use a frame other than my rest frame. Do you disagree?

If you aren't willing to address this question I don't see much point in continuing this conversation.

Also, consider the following quote from p. 43 of https://www.amazon.com/dp/0716723271/?tag=pfamazon01-20 by John Wheeler and Edwin Taylor, about how events (and by extension, worldlines composed of a series of events) are not naturally "in" any particular frame:

'A rocket carries a firecracker. The firecracker explodes. Does this event--the explosion--take place in the rocket frame or in the laboratory frame? Which is the "home" frame for the event? A second firecracker, originally at rest in the laboratory frame, explodes. Does this second event occur in the laboratory frame or in the rocket frame?

'Events are primary, the essential stuff of Nature. Reference frames are secondary, devised by humans for locating and comparing events. A given event occurs in both frames--and in all possible frames moving in all possible directions and with all possible constant relative speeds though the region of spacetime in which the event occurs. The apparatus that "causes" the event may be at rest in one free-float frame; Another apparatus that "causes" a second event may be at rest in a second free-float frame in motion relative to the first. No matter. Each event has its own unique existence. Neither is "owned" by any frame at all.

'A spark jumps 1 millimeter from the antenna of Mary's passing spaceship to a pen in the pocket of John who lounges in the laboratory doorway (Section 1.2). The "apparatus" that makes the spark has parts riding in different reference frames--pen in laboratory frame, antenna in rocket frame. The spark jump--in which frame does this event occur? It is not the property of Mary, not the property of John--not the property of any other observer in the vicinity, no matter what his or her state of motion. The spark-jump event provides data for every observer.

'Drive a steel stake into the ground to mark the corner of a plot of land. Is this a "Daytime stake" or a "Nighttime stake"? Neither! It's just a stake, marking a location in space, the arena of surveying. Similarly an event is neither a "laboratory event" nor a "rocket event." It is just an event, marking a location in spacetime, the arena of science.'

 

[Mike_Fontenot]

[...]
Nothing is stopping B from continuing to use his old inertial rest frame (or any other frame) even when he is no longer at rest in it, there is no reason for him to "transition" unless he chooses to do so!
[...]

The reason for him to "transition", is that if he bases his conclusions about simultaneity on the conclusions of an inertial frame in which he is not at rest, he will then be forced to ignore his own elementary calculations involving his own elementary measurements.

Here is a pertinent post that addresses that issue:

https://www.physicsforums.com/showpost.php?p=3106767&postcount=38 .

The issue that you (JesseM and GrayGhost) are both "dancing around" (the elephant in the room, really), is this: Whenever a person is NOT accelerating for some segment of his life, WHEN in that segment can he legitimately be considered to be an inertial observer? I.e., when is he allowed to use the Lorentz Equations to determine simultaneity at a distance, just as a perpetually-inertial observer is allowed to do?

The correct answer to that question is that he can legitimately be considered to be inertial during the ENTIRE segment. In fact, he MUST be considered to be inertial during the entire segment, in order to maintain consistency with his own elementary calculations.

Other alternatives for the (sometimes accelerating) traveler's simultaneity have been previously endorsed by others on this forum. All of those alternatives answer the above question differently from the answer I gave above ... i.e., they all maintain that a traveler who is not PERPETUALLY inertial CANNOT always use Lorentz simultaneity during segments of their lives in which they are unaccelerated.

For example, Fredrik is a proponent of Dolby&Gull simultaneity:

https://www.physicsforums.com/showpost.php?p=3114946&postcount=16 .

Another alternative, endorsed by Passionflower and Dalespam, is Minguizzi simultaneity:

https://www.physicsforums.com/showpost.php?p=2965909&postcount=72 .

For proponents of these (or any other alternative) simultaneities, EVERY discussion of the standard time-dilation result SHOULD be begun by declaring that neither observer has EVER accelerated in the past. And for some of the alternatives (like Dolby&Gull), it is also necessary to begin by declaring that neither observer will EVER accelerate in the future. Otherwise, those alternatives maintain that the time-dilation result cannot be used at arbitrarily large distances.

Requiring PERPETUALLY-INERTIAL observers, in order to use the Lorentz equations, opens a BIG can-of-worms: Can it really be said (even in principle) that ANY person has NEVER accelerated? We are all composed of elements (and/or their constituents) which have existed since the big bang ... have none of those elements EVER accelerated?

Mike Fontenot

 

[ghwellsjr]

Mike, you don't seem to understand the difference between observers and frames, even after Jesse's very clear description of it.